Processus de Markov étiquetés et systèmes hybrides probabilistes

Tableau d'honneur de la Faculté des études supérieures et postdoctorales, 2011-2012Dans ce mémoire, nous comparons deux modèles de processus probabilistes évoluant dans un environnement continu. Les processus de Markov étiquetés sont des systèmes de transitions pour lesquels l’ensemble des états est non-dénombrable, mais qui évoluent de manière discrète dans le temps. Les mesures de probabilité définies sur l’ensemble des états peuvent avoir un support infini. Les processus hybrides sont une combinaison d’un processus à espace d’états continu qui évolue de manière continue dans le temps et une composante discrète qui intervient pour contrôler l’évolution. Les extensions probabilistes des processus hybrides présentes dans la littérature restreignent le comportement probabiliste à la composante discrète. Nous utilisons deux exemples de systèmes, un avion et un bateau, pour faire ressortir les divergences entre les deux modèles ainsi que leurs limitations, et nous définissons une généralisation qui peut modéliser fidèlement ces exemples. Nous avons également pu montrer, dans un article publié dans un atelier international, comment utiliser, dans le contexte probabiliste, la «substitution d’horloge» et l’«approximation par portrait» qui sont des techniques proposées par Henzinger et al. pour les processus non probabilistes. Ces techniques permettent, sous certaines conditions, de définir un processus probabiliste rectangulaire à partir d’un qui est non rectangulaire, rendant ainsi possible la vérification formelle de toute classe de système hybride probabiliste.We compare two models of processes involving uncountable space. Labelled Markov processes are probabilistic transition systems that can have uncountably many states, but still make discrete time steps. The probability measures on the state space may have uncountable support and a tool has been developed for verification of such systems. Hybrid processes are a combination of a continuous space process that evolves continuously with time and of a discrete component, such as a controller. Existing extensions of Hybrid processes with probability restrict the probabilistic behavior to the discrete component. We have also shown, in a paper, how to compute for probabilistic hybrid systems, the clock approximation and linear phase-portrait approximation that have been proposed for non probabilistic processes by Henzinger et al. The techniques permit, under some conditions, to define a rectangular probabilistic process from a non rectangular one, hence allowing the model-checking of any class of systems. To highlight the differences between Labelled Markov processes and probabilistic hybrid systems, we use two examples, the ones of a boat and an aircraft, an

Stochastic transition systems: bisimulation, logic, and composition

Cyber-physical systems and the Internet of Things raise various challenges concerning the modelling and analysis of large modular systems. Models for such systems typically require uncountable state and action spaces, samplings from continuous distributions, and non-deterministic choices over uncountable many alternatives. In this thesis we fo- cus on a general modelling formalism for stochastic systems called stochastic transition system. We introduce a novel composition operator for stochastic transition systems that is based on couplings of probability measures. Couplings yield a declarative modelling paradigm appropriate for the formalisation of stochastic dependencies that are caused by the interaction of components. Congruence results for our operator with respect to standard notions for simulation and bisimulation are presented for which the challenge is to prove the existence of appropriate couplings. In this context a theory for stochastic transition systems concerning simulation, bisimulation, and trace-distribution relations is developed. We show that under generic Souslin conditions, the simulation preorder is a subset of trace-distribution inclusion and accordingly, bisimulation equivalence is finer than trace-distribution equivalence. We moreover establish characterisations of the simulation preorder and the bisimulation equivalence for a broad subclass of stochastic transition systems in terms of expressive action-based probabilistic logics and show that these characterisations are still maintained by small fragments of these logics, respectively. To treat associated measurability aspects, we rely on methods from descriptive set theory, properties of Souslin sets, as well as prominent measurable-selection principles.:1 Introduction 2 Probability measures on Polish spaces 3 Stochastic transition systems 4 Simulations and trace distributions for Souslin systems 5 Action-based probabilistic temporal logics 6 Parallel composition based on spans and couplings 7 Relations to models from the literature 8 Conclusions 9 Bibliograph

Software components and formal methods from a computational viewpoint

Software components and the methodology of component-based development offer a promising approach to master the design complexity of huge software products because they separate the concerns of software architecture from individual component behavior and allow for reusability of components. In combination with formal methods, the specification of a formal component model of the later software product or system allows for establishing and verifying important system properties in an automatic and convenient way, which positively contributes to the overall correctness of the system. Here, we study such a combined approach. As similar approaches, we also face the so-called state space explosion problem which makes property verification computationally hard. In order to cope with this problem, we derive techniques that are guaranteed to work in polynomial time in the size of the specification of the system under analysis, i.e., we put an emphasis on the computational viewpoint of verification. As a consequence, we consider interesting subclasses of component-based systems that are amenable to such analysis. We are particularly interested in ideas that exploit the compositionality of the component model and refrain from understanding a system as a monolithic block. The assumptions that accompany the set of systems that are verifiable with our techniques can be interpreted as general design rules that forbid to build systems at will in order to gain efficient verification techniques. The compositional nature of software components thereby offers development strategies that lead to systems that are correct by construction. Moreover, this nature also facilitates compositional reduction techniques that allow to reduce a given model to the core that is relevant for verification. We consider properties specified in Computation Tree Logic and put an emphasis on the property of deadlock-freedom. We use the framework of interaction systems as the formal component model, but our results carry over to other formal models for component-based development. We include several examples and evaluate some ideas with respect to experiments with a prototype implementation

Mission and Motion Planning for Multi-robot Systems in Constrained Environments

abstract: As robots become mechanically more capable, they are going to be more and more integrated into our daily lives. Over time, human’s expectation of what the robot capabilities are is getting higher. Therefore, it can be conjectured that often robots will not act as human commanders intended them to do. That is, the users of the robots may have a different point of view from the one the robots do. The first part of this dissertation covers methods that resolve some instances of this mismatch when the mission requirements are expressed in Linear Temporal Logic (LTL) for handling coverage, sequencing, conditions and avoidance. That is, the following general questions are addressed: * What cause of the given mission is unrealizable? * Is there any other feasible mission that is close to the given one? In order to answer these questions, the LTL Revision Problem is applied and it is formulated as a graph search problem. It is shown that in general the problem is NP-Complete. Hence, it is proved that the heuristic algorihtm has 2-approximation bound in some cases. This problem, then, is extended to two different versions: one is for the weighted transition system and another is for the specification under quantitative preference. Next, a follow up question is addressed: * How can an LTL specified mission be scaled up to multiple robots operating in confined environments? The Cooperative Multi-agent Planning Problem is addressed by borrowing a technique from cooperative pathfinding problems in discrete grid environments. Since centralized planning for multi-robot systems is computationally challenging and easily results in state space explosion, a distributed planning approach is provided through agent coupling and de-coupling. In addition, in order to make such robot missions work in the real world, robots should take actions in the continuous physical world. Hence, in the second part of this thesis, the resulting motion planning problems is addressed for non-holonomic robots. That is, it is devoted to autonomous vehicles’ motion planning in challenging environments such as rural, semi-structured roads. This planning problem is solved with an on-the-fly hierarchical approach, using a pre-computed lattice planner. It is also proved that the proposed algorithm guarantees resolution-completeness in such demanding environments. Finally, possible extensions are discussed.Dissertation/ThesisDoctoral Dissertation Computer Science 201

A verification framework for hybrid systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 193-205) and index.Combining; discrete state transitions with differential equations, Hybrid system models provide an expressive formalism for describing software systems that interact with a physical environment. Automatically checking properties, such as invariance and stability, is extremely hard for general hybrid models, and therefore current research focuses on models with restricted expressive power. In this thesis we take a complementary approach by developing proof techniques that are not necessarily automatic, but are applicable to a general class of hybrid systems. Three components of this thesis, namely, (i) semantics for ordinary and probabilistic hybrid models, (ii) methods for proving invariance, stability, and abstraction, and (iii) software tools supporting (i) and (ii), are integrated within a common mathematical framework. (i) For specifying nonprobabilistic hybrid models, we present Structured Hybrid I/O Automata (SHIOAs) which adds control theory-inspired structures, namely state models, to the existing Hybrid I/O Automata, thereby facilitating description of continuous behavior. We introduce a generalization of SHIOAs which allows both nondeterministic and stochastic transitions and develop the trace-based semantics for this framework. (ii) We present two techniques for establishing lower-bounds on average dwell time (ADT) for SHIOA models. This provides a sufficient condition of establishing stability for SHIOAs with stable state models. A new simulation-based technique which is sound for proving ADT-equivalence of SHIOAs is proposed. We develop notions of approximate implementation and corresponding proof techniques for Probabilistic I/O Automata. Specifically, a PIOA A is an E-approximate implementation of B, if every trace distribution of A is c-close to some trace distribution of B-closeness being measured by a metric on the space of trace distributions.(cont.) We present a new class of real-valued simulation functions for proving c-approximate implementations, and demonstrate their utility in quantitatively reasoning about probabilistic safety and termination. (iii) We introduce a specification language for SHIOAs and a theorem prover interface for this language. The latter consists of a translator to typed high order logic and a set of PVS-strategies that partially automate the above verification techniques within the PVS theorem prover.by Sayan Mitra.Ph.D